SUBROUTINE SSYEVR( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
$ ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK,
$ IWORK, LIWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBZ, RANGE, UPLO
INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
REAL ABSTOL, VL, VU
* ..
* .. Array Arguments ..
INTEGER ISUPPZ( * ), IWORK( * )
REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
* ..
*
* Purpose
* =======
*
* SSYEVR computes selected eigenvalues and, optionally, eigenvectors
* of a real symmetric matrix A. Eigenvalues and eigenvectors can be
* selected by specifying either a range of values or a range of
* indices for the desired eigenvalues.
*
* SSYEVR first reduces the matrix A to tridiagonal form T with a call
* to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute
* the eigenspectrum using Relatively Robust Representations. SSTEMR
* computes eigenvalues by the dqds algorithm, while orthogonal
* eigenvectors are computed from various "good" L D L^T representations
* (also known as Relatively Robust Representations). Gram-Schmidt
* orthogonalization is avoided as far as possible. More specifically,
* the various steps of the algorithm are as follows.
*
* For each unreduced block (submatrix) of T,
* (a) Compute T - sigma I = L D L^T, so that L and D
* define all the wanted eigenvalues to high relative accuracy.
* This means that small relative changes in the entries of D and L
* cause only small relative changes in the eigenvalues and
* eigenvectors. The standard (unfactored) representation of the
* tridiagonal matrix T does not have this property in general.
* (b) Compute the eigenvalues to suitable accuracy.
* If the eigenvectors are desired, the algorithm attains full
* accuracy of the computed eigenvalues only right before
* the corresponding vectors have to be computed, see steps c) and d).
* (c) For each cluster of close eigenvalues, select a new
* shift close to the cluster, find a new factorization, and refine
* the shifted eigenvalues to suitable accuracy.
* (d) For each eigenvalue with a large enough relative separation compute
* the corresponding eigenvector by forming a rank revealing twisted
* factorization. Go back to (c) for any clusters that remain.
*
* The desired accuracy of the output can be specified by the input
* parameter ABSTOL.
*
* For more details, see SSTEMR's documentation and:
* - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
* to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
* Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
* - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
* Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
* 2004. Also LAPACK Working Note 154.
* - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
* tridiagonal eigenvalue/eigenvector problem",
* Computer Science Division Technical Report No. UCB/CSD-97-971,
* UC Berkeley, May 1997.
*
*
* Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested
* on machines which conform to the ieee-754 floating point standard.
* SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and
* when partial spectrum requests are made.
*
* Normal execution of SSTEMR may create NaNs and infinities and
* hence may abort due to a floating point exception in environments
* which do not handle NaNs and infinities in the ieee standard default
* manner.
*
* Arguments
* =========
*
* JOBZ (input) CHARACTER*1
* = 'N': Compute eigenvalues only;
* = 'V': Compute eigenvalues and eigenvectors.
*
* RANGE (input) CHARACTER*1
* = 'A': all eigenvalues will be found.
* = 'V': all eigenvalues in the half-open interval (VL,VU]
* will be found.
* = 'I': the IL-th through IU-th eigenvalues will be found.
********** For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and
********** SSTEIN are called
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) REAL array, dimension (LDA, N)
* On entry, the symmetric matrix A. If UPLO = 'U', the
* leading N-by-N upper triangular part of A contains the
* upper triangular part of the matrix A. If UPLO = 'L',
* the leading N-by-N lower triangular part of A contains
* the lower triangular part of the matrix A.
* On exit, the lower triangle (if UPLO='L') or the upper
* triangle (if UPLO='U') of A, including the diagonal, is
* destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* VL (input) REAL
* VU (input) REAL
* If RANGE='V', the lower and upper bounds of the interval to
* be searched for eigenvalues. VL < VU.
* Not referenced if RANGE = 'A' or 'I'.
*
* IL (input) INTEGER
* IU (input) INTEGER
* If RANGE='I', the indices (in ascending order) of the
* smallest and largest eigenvalues to be returned.
* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
* Not referenced if RANGE = 'A' or 'V'.
*
* ABSTOL (input) REAL
* The absolute error tolerance for the eigenvalues.
* An approximate eigenvalue is accepted as converged
* when it is determined to lie in an interval [a,b]
* of width less than or equal to
*
* ABSTOL + EPS * max( |a|,|b| ) ,
*
* where EPS is the machine precision. If ABSTOL is less than
* or equal to zero, then EPS*|T| will be used in its place,
* where |T| is the 1-norm of the tridiagonal matrix obtained
* by reducing A to tridiagonal form.
*
* See "Computing Small Singular Values of Bidiagonal Matrices
* with Guaranteed High Relative Accuracy," by Demmel and
* Kahan, LAPACK Working Note #3.
*
* If high relative accuracy is important, set ABSTOL to
* SLAMCH( 'Safe minimum' ). Doing so will guarantee that
* eigenvalues are computed to high relative accuracy when
* possible in future releases. The current code does not
* make any guarantees about high relative accuracy, but
* future releases will. See J. Barlow and J. Demmel,
* "Computing Accurate Eigensystems of Scaled Diagonally
* Dominant Matrices", LAPACK Working Note #7, for a discussion
* of which matrices define their eigenvalues to high relative
* accuracy.
*
* M (output) INTEGER
* The total number of eigenvalues found. 0 <= M <= N.
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*
* W (output) REAL array, dimension (N)
* The first M elements contain the selected eigenvalues in
* ascending order.
*
* Z (output) REAL array, dimension (LDZ, max(1,M))
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z
* contain the orthonormal eigenvectors of the matrix A
* corresponding to the selected eigenvalues, with the i-th
* column of Z holding the eigenvector associated with W(i).
* If JOBZ = 'N', then Z is not referenced.
* Note: the user must ensure that at least max(1,M) columns are
* supplied in the array Z; if RANGE = 'V', the exact value of M
* is not known in advance and an upper bound must be used.
* Supplying N columns is always safe.
*
* LDZ (input) INTEGER
* The leading dimension of the array Z. LDZ >= 1, and if
* JOBZ = 'V', LDZ >= max(1,N).
*
* ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
* The support of the eigenvectors in Z, i.e., the indices
* indicating the nonzero elements in Z. The i-th eigenvector
* is nonzero only in elements ISUPPZ( 2*i-1 ) through
* ISUPPZ( 2*i ).
********** Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
*
* WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK. LWORK >= max(1,26*N).
* For optimal efficiency, LWORK >= (NB+6)*N,
* where NB is the max of the blocksize for SSYTRD and SORMTR
* returned by ILAENV.
*
* If LWORK = -1, then a workspace query is assumed; the routine
* only calculates the optimal sizes of the WORK and IWORK
* arrays, returns these values as the first entries of the WORK
* and IWORK arrays, and no error message related to LWORK or
* LIWORK is issued by XERBLA.
*
* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
* On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.
*
* LIWORK (input) INTEGER
* The dimension of the array IWORK. LIWORK >= max(1,10*N).
*
* If LIWORK = -1, then a workspace query is assumed; the
* routine only calculates the optimal sizes of the WORK and
* IWORK arrays, returns these values as the first entries of
* the WORK and IWORK arrays, and no error message related to
* LWORK or LIWORK is issued by XERBLA.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: Internal error
*
* Further Details
* ===============
*
* Based on contributions by
* Inderjit Dhillon, IBM Almaden, USA
* Osni Marques, LBNL/NERSC, USA
* Ken Stanley, Computer Science Division, University of
* California at Berkeley, USA
* Jason Riedy, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
$ WANTZ, TRYRAC
CHARACTER ORDER
INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
$ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
$ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
$ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
$ SIGMA, SMLNUM, TMP1, VLL, VUU
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANSY
EXTERNAL LSAME, ILAENV, SLAMCH, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SORMTR, SSCAL, SSTEBZ, SSTEMR, SSTEIN,
$ SSTERF, SSWAP, SSYTRD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
IEEEOK = ILAENV( 10, 'SSYEVR', 'N', 1, 2, 3, 4 )
*
LOWER = LSAME( UPLO, 'L' )
WANTZ = LSAME( JOBZ, 'V' )
ALLEIG = LSAME( RANGE, 'A' )
VALEIG = LSAME( RANGE, 'V' )
INDEIG = LSAME( RANGE, 'I' )
*
LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
*
LWMIN = MAX( 1, 26*N )
LIWMIN = MAX( 1, 10*N )
*
INFO = 0
IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
INFO = -2
ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE
IF( VALEIG ) THEN
IF( N.GT.0 .AND. VU.LE.VL )
$ INFO = -8
ELSE IF( INDEIG ) THEN
IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
INFO = -10
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
INFO = -15
END IF
END IF
*
IF( INFO.EQ.0 ) THEN
NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
NB = MAX( NB, ILAENV( 1, 'SORMTR', UPLO, N, -1, -1, -1 ) )
LWKOPT = MAX( ( NB+1 )*N, LWMIN )
WORK( 1 ) = LWKOPT
IWORK( 1 ) = LIWMIN
*
IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
INFO = -18
ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
INFO = -20
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSYEVR', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
M = 0
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
IF( N.EQ.1 ) THEN
WORK( 1 ) = 26
IF( ALLEIG .OR. INDEIG ) THEN
M = 1
W( 1 ) = A( 1, 1 )
ELSE
IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
M = 1
W( 1 ) = A( 1, 1 )
END IF
END IF
IF( WANTZ )
$ Z( 1, 1 ) = ONE
RETURN
END IF
*
* Get machine constants.
*
SAFMIN = SLAMCH( 'Safe minimum' )
EPS = SLAMCH( 'Precision' )
SMLNUM = SAFMIN / EPS
BIGNUM = ONE / SMLNUM
RMIN = SQRT( SMLNUM )
RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
* Scale matrix to allowable range, if necessary.
*
ISCALE = 0
ABSTLL = ABSTOL
IF (VALEIG) THEN
VLL = VL
VUU = VU
END IF
ANRM = SLANSY( 'M', UPLO, N, A, LDA, WORK )
IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
ISCALE = 1
SIGMA = RMIN / ANRM
ELSE IF( ANRM.GT.RMAX ) THEN
ISCALE = 1
SIGMA = RMAX / ANRM
END IF
IF( ISCALE.EQ.1 ) THEN
IF( LOWER ) THEN
DO 10 J = 1, N
CALL SSCAL( N-J+1, SIGMA, A( J, J ), 1 )
10 CONTINUE
ELSE
DO 20 J = 1, N
CALL SSCAL( J, SIGMA, A( 1, J ), 1 )
20 CONTINUE
END IF
IF( ABSTOL.GT.0 )
$ ABSTLL = ABSTOL*SIGMA
IF( VALEIG ) THEN
VLL = VL*SIGMA
VUU = VU*SIGMA
END IF
END IF
* Initialize indices into workspaces. Note: The IWORK indices are
* used only if SSTERF or SSTEMR fail.
* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
* elementary reflectors used in SSYTRD.
INDTAU = 1
* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
INDD = INDTAU + N
* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
* tridiagonal matrix from SSYTRD.
INDE = INDD + N
* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
* -written by SSTEMR (the SSTERF path copies the diagonal to W).
INDDD = INDE + N
* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
* -written while computing the eigenvalues in SSTERF and SSTEMR.
INDEE = INDDD + N
* INDWK is the starting offset of the left-over workspace, and
* LLWORK is the remaining workspace size.
INDWK = INDEE + N
LLWORK = LWORK - INDWK + 1
* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
* stores the block indices of each of the M<=N eigenvalues.
INDIBL = 1
* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
* stores the starting and finishing indices of each block.
INDISP = INDIBL + N
* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
* that corresponding to eigenvectors that fail to converge in
* SSTEIN. This information is discarded; if any fail, the driver
* returns INFO > 0.
INDIFL = INDISP + N
* INDIWO is the offset of the remaining integer workspace.
INDIWO = INDISP + N
*
* Call SSYTRD to reduce symmetric matrix to tridiagonal form.
*
CALL SSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
$ WORK( INDTAU ), WORK( INDWK ), LLWORK, IINFO )
*
* If all eigenvalues are desired
* then call SSTERF or SSTEMR and SORMTR.
*
TEST = .FALSE.
IF( INDEIG ) THEN
IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
TEST = .TRUE.
END IF
END IF
IF( ( ALLEIG.OR.TEST ) .AND. ( IEEEOK.EQ.1 ) ) THEN
IF( .NOT.WANTZ ) THEN
CALL SCOPY( N, WORK( INDD ), 1, W, 1 )
CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL SSTERF( N, W, WORK( INDEE ), INFO )
ELSE
CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
CALL SCOPY( N, WORK( INDD ), 1, WORK( INDDD ), 1 )
*
IF (ABSTOL .LE. TWO*N*EPS) THEN
TRYRAC = .TRUE.
ELSE
TRYRAC = .FALSE.
END IF
CALL SSTEMR( JOBZ, 'A', N, WORK( INDDD ), WORK( INDEE ),
$ VL, VU, IL, IU, M, W, Z, LDZ, N, ISUPPZ,
$ TRYRAC, WORK( INDWK ), LWORK, IWORK, LIWORK,
$ INFO )
*
*
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by SSTEIN.
*
IF( WANTZ .AND. INFO.EQ.0 ) THEN
INDWKN = INDE
LLWRKN = LWORK - INDWKN + 1
CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA,
$ WORK( INDTAU ), Z, LDZ, WORK( INDWKN ),
$ LLWRKN, IINFO )
END IF
END IF
*
*
IF( INFO.EQ.0 ) THEN
* Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are
* undefined.
M = N
GO TO 30
END IF
INFO = 0
END IF
*
* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
* Also call SSTEBZ and SSTEIN if SSTEMR fails.
*
IF( WANTZ ) THEN
ORDER = 'B'
ELSE
ORDER = 'E'
END IF
CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
$ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWK ),
$ IWORK( INDIWO ), INFO )
*
IF( WANTZ ) THEN
CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
$ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
$ WORK( INDWK ), IWORK( INDIWO ), IWORK( INDIFL ),
$ INFO )
*
* Apply orthogonal matrix used in reduction to tridiagonal
* form to eigenvectors returned by SSTEIN.
*
INDWKN = INDE
LLWRKN = LWORK - INDWKN + 1
CALL SORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
$ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
END IF
*
* If matrix was scaled, then rescale eigenvalues appropriately.
*
* Jump here if SSTEMR/SSTEIN succeeded.
30 CONTINUE
IF( ISCALE.EQ.1 ) THEN
IF( INFO.EQ.0 ) THEN
IMAX = M
ELSE
IMAX = INFO - 1
END IF
CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
END IF
*
* If eigenvalues are not in order, then sort them, along with
* eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
* It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do
* not return this detailed information to the user.
*
IF( WANTZ ) THEN
DO 50 J = 1, M - 1
I = 0
TMP1 = W( J )
DO 40 JJ = J + 1, M
IF( W( JJ ).LT.TMP1 ) THEN
I = JJ
TMP1 = W( JJ )
END IF
40 CONTINUE
*
IF( I.NE.0 ) THEN
W( I ) = W( J )
W( J ) = TMP1
CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
END IF
50 CONTINUE
END IF
*
* Set WORK(1) to optimal workspace size.
*
WORK( 1 ) = LWKOPT
IWORK( 1 ) = LIWMIN
*
RETURN
*
* End of SSYEVR
*
END